Fig. 1.

(A) Polarized optical micrographs of a single Saturn-ring structure, an entangled dimer, and an entangled cluster of particles with a homeotropic surface anchoring in a right-handed $90°$ TN cell. (B) Side view of the director field and disclinations in the Saturn-ring structure from A showing the effect of a $90°$ twist. (Inset) Direction of view. (C) Sketch of the director streamlines around the dimer in a horizontal midplane cross-section. Notice the disclinations delimit opposite twist domains with perpendicular midplane director orientation. (D) Example of disclinations entangling a $4\times 3$ array of particles. (E) Idealized and rectangularly aligned depiction of a knot diagram for the structure in D. The tangles between four adjacent particles can exist in three different states, which can be individually switched with laser tweezers. (F) Topology-preserving simplification of the disclination geometry in E identifies a figure-eight knot ($4_1$).